For proving the \(C^\infty \)  equivalence between \(\widetilde{\mathcal {V}}\)  and \(\widetilde{\mathcal {V}}+R^-\) , we adopt the homological method (see e.g. [1]} or [2]}). Set
\(\mathcal {V}_s=\widetilde{\mathcal {V}}+s R^-,\quad s\in [0,1].\) 
